Regularity Of Minimizers Research Articles

Regularity for minimizers of scalar integral functionals with (p, q)-growth conditions

Regularity for minimizers of scalar integral functionals with (p, q)-growth conditions

Quasiconvex Functionals of (p, q)-Growth and the Partial Regularity of Relaxed Minimizers

We establish C∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{C}^{\\infty }$$\\end{document}-partial regularity results for relaxed minimizers of strongly quasiconvex functionals F[u;Ω]:=∫ΩF(∇u)dx,u:Ω→RN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathscr {F}[u;\\Omega ]:=\\int _{\\Omega }F(\ abla u)\ extrm{d}x,\\qquad u:\\Omega \\rightarrow \\mathbb {R}^{N}, \\end{aligned}$$\\end{document}subject to a q-growth condition |F(z)|≦c(1+|z|q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|F(z)|\\leqq c(1+|z|^{q})$$\\end{document}, z∈RN×n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z\\in \\mathbb {R}^{N\ imes n}$$\\end{document}, and natural p-mean coercivity conditions on F∈C∞(RN×n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F\\in \ extrm{C}^{\\infty }(\\mathbb {R}^{N\ imes n})$$\\end{document} for the basically optimal exponent range 1≦p≦q<min{npn-1,p+1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\leqq p\\leqq q<\\min \\{\\frac{np}{n-1},p+1\\}$$\\end{document}. With the p-mean coercivity condition being stated in terms of a strong quasiconvexity condition on F, our results include pointwise (p, q)-growth conditions as special cases. Moreover, we directly allow for signed integrands which is natural in view of coercivity considerations and hence the direct method, but is novel in the study of relaxed problems. In the particular case of classical pointwise (p, q)-growth conditions, our results extend the previously known exponent range from Schmidt’s foundational work (Schmidt in Arch Ration Mech Anal 193:311–337, 2009) for non-negative integrands to the maximal range for which relaxations are meaningful, moreover allowing for p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document}. We also emphasize that our results apply to the canonical class of signed integrands and do not rely in any way on measure representations à la Fonseca and Malý (Ann Inst Henri Poincaré Anal Non Linéaire 14:309–338, 1997).

Ahlfors regularity of continua that minimize maxitive set functions

The primary objective of this paper is to establish the Ahlfors regularity of minimizers of set functions that satisfy a suitable maxitive condition on disjoint unions of sets. Our analysis focuses on minimizers within continua of the plane with finite 1-dimensional Hausdorff measure. Through quantitative estimates, we prove that the length of a minimizer inside the ball centered at one of its points is comparable to the radius of the ball. By operating within an abstract framework, we are able to encompass a diverse range of entities, including the inradius of a set, the maximum of the torsion function, and spectral functionals defined in terms of the eigenvalues of elliptic operators. These entities are of interest for several applications, such as structural engineering, urban planning, and quantum mechanics.

Quantified Legendreness and the Regularity of Minima

We introduce a new quantification of nonuniform ellipticity in variational problems via convex duality, and prove higher differentiability and 2d-smoothness results for vector valued minimizers of possibly degenerate functionals. Our framework covers convex, anisotropic polynomials as prototypical model examples—in particular, we improve in an essentially optimal fashion Marcellini’s original results (Marcellini in Arch Rat Mech Anal 105:267–284, 1989).

Thomas–Fermi theory of out-of-plane charge screening in graphene

This paper provides a variational treatment of the effect of external charges on the free charges in an infinite free-standing graphene sheet within the Thomas–Fermi theory. We establish existence, uniqueness and regularity of the energy minimizers corresponding to the free charge densities that screen the effect of an external electrostatic potential at the neutrality point. For the potential due to one or several off-layer point charges, we also prove positivity and a precise universal asymptotic decay rate for the screening charge density, as well as an exact charge cancellation by the graphene sheet. We also treat a simpler case of the non-zero background charge density and establish similar results in that case.

Some remarks on the higher regularity of minimizers of anisotropic functionals

Some remarks on the higher regularity of minimizers of anisotropic functionals

On the N-Cheeger problem for component-wise increasing norms

We study Cheeger and p-eigenvalue partition problems depending on a given evaluation function Φ for p∈[1,∞). We prove existence and regularity of minima, relations between the problems, convergence, and stability with respect to p and to Φ.

Partial regularity for minimizers of a class of discontinuous Lagrangians

We study a one-dimensional Lagrangian problem including the variational reformulation, derived in a recent work of Ambrosio–Baradat–Brenier, of the discrete Monge–Ampère gravitational model, which describes the motion of interacting particles whose dynamics is ruled by the optimal transport problem. The more general action-type functional we consider contains a discontinuous potential term related to the descending slope of the opposite squared distance function from a generic discrete set in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^{d}$$\\end{document}. We exploit the underlying geometrical structure provided by the associated Voronoi decomposition of the space to obtain C1,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,1}$$\\end{document}-regularity for local minimizers out of a finite number of shock times.

Direct Minimization of the Canham–Helfrich Energy on Generalized Gauss Graphs

The existence of minimizers of the Canham–Helfrich functional in the setting of generalized Gauss graphs is proved. As a first step, the Canham–Helfrich functional, usually defined on regular surfaces, is extended to generalized Gauss graphs, then lower semicontinuity and compactness are proved under a suitable condition on the bending constants ensuring coerciveness; the minimization follows by the direct methods of the Calculus of Variations. Remarks on the regularity of minimizers and on the behavior of the functional in case there is lack of coerciveness are presented.

Higher regularity for minimizers of very degenerate convex integrals

In this paper, we consider minimizers of integral functionals of the type F(u)≔∫Ω1p(|Du(x)|γ(x)−1)+pdx,for p>1, where u:Ω⊂Rn→RN, with N≥1, is a possibly vector-valued function. Here, |⋅|γ is the associated norm of a bounded, symmetric and coercive bilinear form on RnN. We prove that K(x,Du) is continuous in Ω, for any continuous function K:Ω×RnN→R vanishing on {(x,ξ)∈Ω×RnN:|ξ|γ(x)≤1}.

Partial regularity of minimizers for double phase functionals with variable exponents

Partial regularity of minimizers for double phase functionals with variable exponents

Torus-like solutions for the Landau-de Gennes model. Part II: Topology of [formula omitted]-equivariant minimizers

We study energy minimization of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in a three-dimensional axisymmetric domain and in a restricted class of S1-equivariant (i.e., axially symmetric) configurations. We assume smooth and nonvanishing S1-equivariant (e.g. homeotropic) Dirichlet boundary condition and a physically relevant norm constraint (the Lyuksyutov constraint) in the interior. Relying on our previous results in the nonsymmetric setting [16], we prove partial regularity of minimizers away from a possible finite set of interior singularities located on the symmetry axis. For a suitable class of domains and boundary data, we show that for smooth minimizers (torus solutions) the level sets of the signed biaxiality are generically finite unions of tori of revolution. Concerning nonsmooth minimizers (split solutions), we characterize their asymptotic behavior around any singular point in terms of explicit S1-equivariant harmonic maps into S4, whence the generic level sets of the signed biaxiality contains invariant topological spheres. Finally, in the model case of a nematic droplet, we provide existence of torus solutions, at least when the boundary data are suitable uniaxial deformations of the radial anchoring, and existence of split solutions for boundary data which are suitable linearly full harmonic spheres.

A vectorial problem with thin free boundary

We consider the vectorial analogue of the thin free boundary problem introduced by Caffarelli et al. (J Eur math Soc 12:1151-1179, 2010) as a realization of a nonlocal version of the classical Bernoulli problem. We study optimal regularity, nondegeneracy, and density properties of local minimizers. Via a blow-up analysis based on a Weiss type monotonicity formula, we show that the free boundary is the union of a “regular” and a “singular” part. Finally we use a viscosity approach to prove C^{1,alpha } regularity of the regular part of the free boundary.

The Alt–Phillips functional for negative powers

AbstractWe develop the free boundary regularity for nonnegative minimizers of the Alt–Phillips functional for negative power potentials and establish a ‐convergence result of the rescaled energies to the perimeter functional as .

A Remark on Tonelli’s Calculus of Variations

This paper provides a quite simple method of Tonelli’s calculus of variations with positive definite and superlinear Lagrangians. The result complements the classical literature of calculus of variations before Tonelli’s modern approach. Inspired by Euler’s spirit, the proposed method employs finite-dimensional approximation of the exact action functional, whose minimizer is easily found as a solution of Euler’s discretization of the exact Euler – Lagrange equation. The Euler – Cauchy polygonal line generated by the approximate minimizer converges to an exact smooth minimizing curve. This framework yields an elementary proof of the existence and regularity of minimizers within the family of smooth curves and hence, with a minor additional step, within the family of Lipschitz curves, without using modern functional analysis on absolutely continuous curves and lower semicontinuity of action functionals.

Regularity of minimizers for free-discontinuity problems withp(·)-growth

A regularity result for free-discontinuity energies defined on the spaceSBVp(·)of special functions of bounded variation with variable exponent is proved, under the assumption of a log-Hölder continuity for the variable exponentp(x). Our analysis expands on the regularity theory for minimizers of a class of free-discontinuity problems in the nonstandard growth case. This may be seen as a follow-up of the paper N. Fusco et al.,J. Convex Anal.8 (2001) 349-367, dealing with a constant exponent.

Well-posedness and regularity for a polyconvex energy

We prove the existence, uniqueness, and regularity of minimizers of a polyconvex functional in two and three dimensions, which corresponds to the H1-projection of measure-preserving maps. Our result introduces a new criteria on the uniqueness of the minimizer, based on the smallness of the lagrange multiplier. No estimate on the second derivatives of the pressure is needed to get a unique global minimizer. As an application, we construct a minimizing movement scheme to construct Lr-solutions of the Navier–Stokes equation (NSE) for a short time interval.

Lipschitz regularity of minimizers of variational integrals with variable exponents

In this paper we prove the Lipschitz regularity for local minimizers of convex variational integrals of the form F(v,Ω)=∫ΩF(x,Dv(x))dx,where, for n≥2 and N≥1, Ω is a bounded open set in Rn, u∈W1,1(Ω,RN) and the energy density F:Ω×RN×n→R satisfies the so called variable growth conditions. The main novelty here is that we assume an almost critical regularity in the Orlicz Sobolev setting for the energy density as a function of the x variable.

Regularity of Minimizers for a Model of Charged Droplets

We investigate properties of minimizers of a variational model describing the shape of charged liquid droplets. The model, proposed by Muratov and Novaga, takes into account the regularizing effect due to the screening of free counterionions in the droplet. In particular we prove partial regularity of minimizers, a first step toward the understanding of further properties of minimizers.

Optimal regularity for variational solutions of free transmission problems

In this article we study functionals of the type considered in [36], i.e.J(v):=∫B1(A(x,u)|∇u|2+f(x,u)u+Q(x)λ(u))dx here A(x,u)=A+(x)χ{u>0}+A−(x)χ{u<0}, f(x,u)=f+(x)χ{u>0}+f−(x)χ{u<0} and λ(x,u)=λ+(x)χ{u>0}+λ−(x)χ{u≤0}. We prove the optimal C0,1− regularity of minimizers of the functional indicated above (with precise estimates) when the coefficients A± are continuous functions and μ≤A±≤1μ for some 0<μ<1, with f∈LN(B1) and Q bounded. We do this by presenting a new compactness argument and approximation theory similar to the one developed by L. Caffarelli in [9] to treat the regularity theory for solutions to fully nonlinear PDEs. Moreover, we introduce the Ta,b operator that allows one to transfer minimizers from the transmission problems to the Alt-Caffarelli-Friedman type functionals, in small scales, allowing this way the study of the regularity theory of minimizers of Bernoulli type free transmission problems.